New directions in categorical logic, for classical, probabilistic and quantum logic

Abstract

Intuitionistic logic, in which the double negation law ¬¬P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold, especially in quantitative logics for probabilistic and quantum systems. The algebraic notions of effect algebra and effect module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X → 1 + 1 carry such effect module structure, and can be used as predicates. Maps of this form X → 1 + 1 are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C∗-algebra or Hilbert space. In quantum foundations the duality between states and effects (predicates) plays an important role. This duality appears in the form of an adjunction in our categorical setting, where we use maps 1 → X as states. For such a state ω and a predicate p, the validity probability ω |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of ‘instruments’, using Lüders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C∗-algebras, side-effects appear exclusively in the non- commutative (properly quantum) case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems, in which the above mentioned elements occur.